We are now in position to prove the first part of Lemma .
[Proof of Lemma ]
We first observe that, via the variational definition of entropy and the fact that \(M_{m} \subset M_{m+1}\) , for all \(m \in \mathbb {N}\) , we obtain that \(H(\upsilon ^m|\mu ^m)\)  is monotone increasing in \(m\)  (following the steps pointed out in Remark  or directly as a consequence of [1]}). In particular,
\(\sup _m H(\upsilon ^m|\mu ^m)=\lim _{m \rightarrow \infty } H(\upsilon ^m|\mu ^m)\) 
